Question

in an AP the sum of first n terms is 3 n²/2+13n/2. find this 25th term

Answer 1

$The\ sum\ of\ first\ n\ terms\ of\ any\ AP =S_n = \frac{d}{2}n^2 + (a - \frac{d}{2})n. \\ \\ a = first\ term \\ \\ d = common\ difference$

$S_n = \frac{3n^2}{2} + \frac{13n}{2} \\ \\ = \frac{3}{2}n^2 + \frac{13}{2}n \\ \\ \\ Co-efficient\ of\ n^2 = \frac{d}{2} = \frac{3}{2} \\ \\ d = \frac{3}{2} \times 2 = 3 \\ \\ \\ Co-efficient\ of\ n = a - \frac{d}{2} = \frac{13}{2} \\ \\ = a - \frac{3}{2} = \frac{13}{2} \\ \\ a = \frac{13}{2} + \frac{3}{2} = \frac{16}{2} = 8 \\ \\ \\ T_{25} = a + 24d \\ \\ = 8 + 24 \times 3 \\ \\ = 8 + 72 = 80 \\ \\ \\ 80\ is\ the\ answer.$

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